An immersive visual representation depicting a conic section related to the equation 4x^2+9y^2=36. Illustrate this by drawing a clear graph showing the conic shape, the x and y axes, and the defining intersection points that determine its shape. Please ensure the image contains no text elements.

What type of conic section is given by the equation 4x^2+9y^2=36? What are its domain and range?

It'll be an ellipse.

The domain -3, 3.
The range -2 2.

Am I right?

yes. good work

Yeah thanx

Oh, I see we're getting into some math now! That equation represents an ellipse. As for its domain and range, well, let's just say this ellipse has no boundaries—it's as big as your love for ice cream, infinite if you will! So grab a scoop and enjoy the infinite sweet goodness of math!

To find the type of conic section and its domain and range, we need to analyze the given equation:

4x^2 + 9y^2 = 36

By comparing this equation to the general form of conic sections, it resembles the equation of an ellipse. An ellipse is a conic section where the sum of the squares of the x-coefficients and y-coefficients is less than one. In this case, both 4 and 9 are less than 1.

Hence, the given equation represents an ellipse.

To determine the domain and range of the ellipse, we need to consider which values x and y can take.

For the variable x, it can take any real value because there are no restrictions on it in the equation. So, the domain of the ellipse is (-∞, +∞).

For the variable y, we can determine its range by isolating it in the equation:

4x^2 + 9y^2 = 36

Dividing both sides by 36:

x^2/9 + y^2/4 = 1

Now we can find the square root of both sides:

√(x^2/9) + √(y^2/4) = √1

x/3 + y/2 = 1

Rearranging the equation:

y/2 = -x/3 + 1

y = -2x/3 + 2

From this equation, we can see that y is a linear function of x. It means that y can take any real value, as there are no restrictions. So, the range of the ellipse is (-∞, +∞).

In summary:
- The type of conic section given by the equation is an ellipse.
- The domain of the ellipse is (-∞, +∞).
- The range of the ellipse is (-∞, +∞).

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